Game Development Reference
In-Depth Information
Before we look at some other properties of
J
, let's finish extending the basic
formulas in three dimensions with perhaps the most important one.
In the plane, the rotational equivalent of Newton's second law, F = ma,
is τ = Jα. Extending this into three dimensions is straightforward.
3D Rotational Analog of Newton's Second Law
τ = α
J
.
As we've said before, in computer simulations it's often the case that the
force and mass are known, and we compute the acceleration using a = F/m.
A similar situation exists for rotational dynamics, where division by m is
replaced with multiplication by
J
−1
:
−1
.
α = τ
J
As it turns out, the inverse inertia tensor is needed more frequently in
digital simulations than is
J
, and it is usually precomputed and stored.
Equation (12.30) tells us how to compute the inertia tensor for a point
mass, but what about more complicated shapes? In a manner similar to
how we computed the center of mass, we can imagine breaking up a com-
pound object into a large number of mass elements and taking the sum
of their individual moments of inertia. Taking the limit as the volume of
the largest element approaches zero, this sum becomes a multidimensional
integral. Such integrals are typically di
cult or impossible to solve analyti-
cally, except for abstract primitives such as boxes, disks, cylinders, spheres,
cones, and the like. Fortunately, such primitives arise commonly in prac-
tice and can make adequate approximations. Even more fortunate for us is
the fact that the hard work of solving the integral has already been done
for a large variety of primitives. For such primitives, the best method for
obtaining the moment of inertia is to look up the formula in a table. (At
the time of this writing, such a table can be found on wikipedia.org under
“List of moment of inertia tensors.”)
More complicated objects are typically approximated by breaking down
the object into primitives with known formulas, calculating their moments
matrices encountered in 3D interactive applications are transformation matrices, and the
left-to-right reading order is a useful advantage. In the derivation above, we used column
vectors for pedagogical and aesthetic purposes (the inertia tensor is not a transform
matrix), but shame on us for being inconsistent. From here on out, we'll stick with our
convention and put the vectors on the left.
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